Polygon trimming using a modified azimuthal map projection

Abstract

In one embodiment, a system, method and computer readable media are disclosed for trimming geographic data that defines polygons to boundaries of a quadrangle. The geographic data is converted to map coordinates in a novel azimuthal map projection. Each map projected point in the azimuthal map projection has an azimuthal angle that is proportional to longitude of the point, and has a radius from the center of the azimuthal map projection that is proportional to both the longitude and latitude of the point. Polygons are trimmed along a lower limit of the quadrangle and along an upper longitude limit of the quadrangle using the azimuthal map projection. The map coordinates in the azimuthal map projection are then converted back to geographic data.

1. A method for trimming geographic data that defines polygons to boundaries of a quadrangle, comprising:
 converting the geographic data into map coordinates in a modified azimuthal map projection, a map coordinate in the modified azimuthal map projection describing a projected point having an azimuthal angle proportional to longitude of the point, and having a radius from the center of the modified azimuthal map projection proportional to both longitude and latitude of the point such that parallels in the modified azimuthal map projection are projected as squares, the modified azimuthal map projection being non-equidistant; 
 trimming one or more polygons along a lower longitude limit of the quadrangle in the modified azimuthal map projection; 
 trimming one or more polygons along an upper longitude limit of the quadrangle in the modified azimuthal map projection; and 
 converting the map coordinates in the modified azimuthal map projection back to geographic data. 
 
     
     
       2. The method of  claim 1 , wherein the modified azimuthal map projection is a polar modified azimuthal map projection. 
     
     
       3. The method of  claim 1 , further comprising:
 determining if a lower latitude limit of the quadrangle is within a threshold distance of a first point and an upper latitude limit is not within the threshold distance of a second point, and if so, using the first point as a center of the modified azimuthal map projection. 
 
     
     
       4. The method of  claim 3 , further comprising:
 determining if an upper latitude limit of the quadrangle is within the threshold distance of a second point and a lower latitude limit is not within the threshold distance of the first point, and if so, using the second point as a center of the modified azimuthal map projection. 
 
     
     
       5. The method of  claim 4 , wherein the first point is the South Pole and the second point is the North Pole. 
     
     
       6. The method of  claim 1 , further comprising:
 determining if a lower latitude limit of the quadrangle is within a threshold distance of a first point and an upper latitude limit is within the threshold distance of a second point, and if so, segmenting the geographic data into a first portion and a second portion; and 
 applying the first portion to the step of converting the geographic data into map coordinates in a modified azimuthal map projection. 
 
     
     
       7. The method of  claim 1 , wherein trimming polygons along a lower longitude limit further comprises:
 rotating the modified azimuthal map projection so that the lower longitude limit coincides with a vertical line; and 
 trimming polygons along the vertical line to remove portions to a first side of the vertical line. 
 
     
     
       8. The method of  claim 6 , wherein trimming polygons along an upper longitude limit further comprises:
 rotating the modified azimuthal map projection so that the upper longitude limit coincides with the vertical line; and 
 trimming polygons along the vertical line to remove portions to a second side of the vertical line. 
 
     
     
       9. The method of  claim 1 , wherein the geographic data comprises a latitude vector that includes latitudes of vertices of polygons, and a longitude vector that includes longitudes of vertices of polygons. 
     
     
       10. The method of  claim 1 , wherein converting the geographic data into map coordinates in a modified azimuthal map projection further comprises:
 setting an x-axis value of a projected point using calculation of sine of longitude of the point; and 
 setting a y-axis value of a projected point using calculation of cosine of longitude of the point. 
 
     
     
       11. The method of  claim 9 , further comprising:
 multiplying sine of the longitude of the point by the sum of latitude of the point and a constant divided by an adjustment factor; and 
 multiplying cosine of the longitude of the point by the sum of latitude of the point and the constant divided by the adjustment factor. 
 
     
     
       12. The method of  claim 1 , wherein converting the geographic data into map coordinates in a modified azimuthal map projection further comprises:
 employing the expressions,
     r =(lat+pi/2)/scalefactor(lon); 
     x=r *sin(−lon);
 
     y=r *cos(lon); 
 
 
       where “lat” and “lon” are latitude and longitude vectors storing polygon data, “scalefactor(lon)” is an adjustment factor based upon longitude, “x” and “y” are vectors storing x-axis and y-axis values of map coordinates of polygon vertices in the modified azimuthal map projection, and “pi” is an approximation of the mathematical constant pi. 
     
     
       13. An apparatus for trimming geographic data that defines polygons to boundaries of a quadrangle, comprising:
 a processor configured to execute instructions for one or more units; and 
 a polygon to quadrangle trimming unit configured to convert the geographic data into map coordinates in a modified azimuthal map projection, a map coordinate in the modified azimuthal map projection describing a projected point having an azimuthal angle proportional to longitude of the point, and having a radius from the center of the modified azimuthal map projection proportional to both longitude and latitude of the point such that parallels in the modified azimuthal map projection are projected as squares, wherein the modified azimuthal map projection is non-equidistant, 
 wherein the polygon to quadrangle trimming unit is further configured to trim polygons along a lower and an upper longitude limit of the quadrangle in the modified azimuthal map projection, and to convert the map coordinates in the modified azimuthal map projection back to geographic data. 
 
     
     
       14. The apparatus of  claim 13 , wherein the modified azimuthal map projection is a polar modified azimuthal map projection. 
     
     
       15. The apparatus of  claim 13 , wherein the polygon to quadrangle trimming unit further comprises:
 a decision unit configured to determine if a lower latitude limit of the quadrangle is within a threshold distance of a first point and if an upper latitude limit is not within the threshold distance of a second point, and if so, to cause the polygon to quadrangle trimming unit to use the first point as a center of the modified azimuthal map projection. 
 
     
     
       16. The apparatus of  claim 15 , wherein the decision unit is further configured to determine if an upper latitude limit of the quadrangle is within a threshold distance of a second point and a lower latitude limit is not within the threshold distance of the first point, and if so, to cause the polygon to quadrangle trimming unit to use the second point as a center of the modified azimuthal map projection. 
     
     
       17. The apparatus of  claim 16 , wherein the first point is the South Pole and the second point is the North Pole. 
     
     
       18. The apparatus of  claim 13 , wherein the polygon to quadrangle trimming unit further comprises:
 a decision unit configured to determine if a lower latitude limit of the quadrangle is within a threshold distance of a first point and an upper latitude limit is within the threshold distance of a second point, and if so, to cause the polygon to quadrangle trimming unit to segment the geographic data into a first portion and a second portion, and to convert the first portion into map coordinates in the modified azimuthal map projection. 
 
     
     
       19. The apparatus of  claim 13 , wherein the polygon to quadrangle trimming unit is further configured to rotate the modified azimuthal map projection so that the lower longitude limit coincides with a vertical line and to trim polygons along the vertical line to remove portions to a first side of the vertical line. 
     
     
       20. The apparatus of  claim 19 , wherein the polygon to quadrangle trimming unit is further configured to rotate the modified azimuthal map projection so that the upper longitude limit coincides with the vertical line, and to trim polygons along the vertical line to remove portions to a second side of the vertical line. 
     
     
       21. The apparatus of  claim 13 , wherein the geographic data comprises a latitude vector that includes latitudes of vertices of polygons, and a longitude vector that includes longitudes of vertices of polygons. 
     
     
       22. The apparatus of  claim 13  wherein the polygon to quadrangle trimming unit is further configured to set an x-axis value of a projected point using calculation of sine of longitude of the point, and to set a y-axis value of a projected point using calculation of cosine of longitude of the point. 
     
     
       23. The apparatus of  22 , wherein the polygon to quadrangle trimming unit is further configured to multiply sine of the longitude of the point by the sum of latitude of the point and a constant, divided by an adjustment factor, and multiply cosine of the longitude of the point by the sum of latitude of the point and the constant, divided by the adjustment factor. 
     
     
       24. The apparatus of  13 , wherein the polygon to quadrangle trimming unit is further configured to employ the expressions,
     r =(lat+pi/2)/scalefactor(lon); 
     x=r *sin(−lon);
 
     y=r *cos(lon); 
 
       where “lat” and “lon” are latitude and longitude vectors storing polygon data, “scalefactor(lon)” is an adjustment factor based upon longitude, “x” and “y” are vectors storing x-axis and y-axis values of map coordinates of polygon vertices in the modified azimuthal map projection, and “pi” is an approximation of the mathematical constant pi. 
     
     
       25. An apparatus for trimming geographic data to boundaries of a quadrangle, comprising:
 means for converting the geographic data into map coordinates in a modified azimuthal map projection, a map coordinate in the modified azimuthal map projection to describe a projected point having an azimuthal angle proportional to longitude of the point, and having a radius from the center of the modified azimuthal map projection proportional to both longitude and latitude of the point such that parallels in the modified azimuthal map projection are projected as squares, wherein the modified azimuthal map projection is non-equidistant; 
 means for trimming one or more polygons along a lower longitude limit of the quadrangle in the modified azimuthal map projection; 
 means for trimming the one or more polygons along an upper longitude limit of the quadrangle in the modified azimuthal map projection; and 
 means for converting the map coordinates in the modified azimuthal map projection back to geographic data. 
 
     
     
       26. A non-transitory, tangible computer-readable media comprising program instructions that, when executed, are operable to:
 convert geographic data into map coordinates in a modified azimuthal map projection, a map coordinate in the modified azimuthal map projection to describe a projected point having an azimuthal angle proportional to longitude of the point, and having a radius from the center of the modified azimuthal map projection proportional to both longitude and latitude of the point such that parallels in the modified azimuthal map projection are projected as squares, wherein the modified azimuthal map projection is non-equidistant; 
 trim one or more polygons along a lower longitude limit of the quadrangle in the modified azimuthal map projection; 
 trim one or more polygons along an upper longitude limit of the quadrangle in the modified azimuthal map projection; and 
 convert the map coordinates in the modified azimuthal map projection back to geographic data. 
 
     
     
       27. The computer-readable media of  claim 26 , wherein the modified azimuthal map projection is a polar modified azimuthal map projection. 
     
     
       28. The computer-readable media of  claim 26 , further comprising program instructions that, when executed, are operable to:
 determine if a lower latitude limit of the quadrangle is within a threshold distance of a first point and an upper latitude limit is not within the threshold distance of a second point, and if so, use the first point as a center of the modified azimuthal map projection. 
 
     
     
       29. The computer-readable media of  claim 28 , wherein the first point is the South Pole. 
     
     
       30. The computer-readable media of  claim 26 , wherein the geographic data comprises a latitude vector that includes latitudes of vertices of polygons, and a longitude vector that includes longitudes of vertices of polygons. 
     
     
       31. The computer-readable media of  claim 26 , further comprising program instructions that, when executed, are operable to:
 set an x-axis value of a projected point using calculation of sine of longitude of the point; and 
 set a y-axis value of a projected point using calculation of cosine of longitude of the point. 
 
     
     
       32. The computer-readable media of  claim 26 , further comprising program instructions that, when executed, are operable to:
 multiply sine of the longitude of the point by the sum of latitude of the point and a constant divided by an adjustment factor; and 
 multiply cosine of the longitude of the point by the sum of latitude of the point and the constant divided by the adjustment factor. 
 
     
     
       33. The computer-readable media of  claim 26 , further comprising program instructions that, when executed, are operable to:
 employ the expressions,
     r =(lat+pi/2)/scalefactor(lon); 
     x=r *sin(−lon);
 
     y=r *cos(lon); 
 
 
       where “lat” and “lon” are latitude and longitude vectors storing polygon data, “scalefactor(lon)” is an adjustment factor based upon longitude, “x” and “y” are vectors storing x-axis and y-axis values of map coordinates of polygon vertices in the modified azimuthal map projection, and “pi” is an approximation of the mathematical constant pi.